Collaborative Research: A Posteriori Error Analysis for Complex Models with Applications to Efficient Numerical Solution and Uncertainty Quantification

协作研究：复杂模型的后验误差分析及其在高效数值求解和不确定性量化中的应用

基本信息

批准号：
1720402
负责人：
Jehanzeb Chaudhary
金额：
$ 10.02万
依托单位：
University of New Mexico
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720402&HistoricalAwards=false
关键词：
Collaborative Research Posteriori Error Analysis

项目摘要

Many scientific and engineering problems of importance to the nation's infrastructure and defense are concerned with multi-physics systems in which multiple physical processes interact in complex ways. An important example is the flow of a liquid transporting reacting chemicals in which the reaction affects the fluid properties of the liquid. Such reacting flows arise in applications ranging from biological systems to combustion processes associated with energy use. In general, the complexity of multi-physics systems prevents direct experimental observation of crucial features. Thus their study depends critically on computing approximate solutions of mathematical models describing the processes and their interactions. However, such simulations strain the computational capabilities of the most powerful computers and consequently, computational errors in the approximations are always significant and may be overwhelming. Over two decades, the project investigators have developed a systematic approach for producing accurate computational estimates of the error of approximate solutions of models of multi-physics systems. In this project, the investigators explore the use of error estimates from this approach to guide the efficient use of computational resources in order to maximize the fidelity of approximate solutions of multi-physics systems. They also apply the error estimates to accurately quantify the uncertainty in predictions of behavior of multi-physics systems based on the approximate solutions of models. The results of this project will enhance the ability of the nation's engineers and scientists to investigate and predict the behavior of complex physical systems important to the nation's security and infrastructure. This project tackles critical problems associated with using sophisticated cutting-edge multi-discretization numerical methods for multiscale, multiphysics models to pursue scientific inference and engineering design. The research is based on a posteriori error analysis for multi-physics, multi-discretization problems that quantifies the effects of a wide variety of discretization steps through the use of adjoint problems and computable residuals. The primary focus of the project is twofold: (1) Developing and analyzing methods for using accurate error estimates to guide discretization choices in order to achieve a desired accuracy at roughly minimal computational cost; and (2) Investigating how to extend accurate error estimation methods to address uncertainty quantification for multiphysics systems, where 'discretization' includes the sampling of a random process and the overall error is a combination of discretization and sampling errors. The investigators pursue the development of novel multi-stage approaches to the construction of efficient numerical solutions and the extension of a posteriori error analysis to statistical computations. The project also involves the extension of the theory of a posteriori error analysis to hyperbolic problems and nonstandard quantities of interest.

许多对国家基础设施和防御重要性的科学和工程问题与多物理系统有关，其中多种物理过程以复杂的方式相互作用。一个重要的例子是转运化学物质的液体的流动，其中反应影响液体的流体特性。这种反应流在从生物系统到与能源使用相关的燃烧过程等应用中出现。通常，多物理系统的复杂性阻止了对关键特征的直接实验观察。因此，他们的研究严格取决于计算描述过程及其相互作用的数学模型的近似解。但是，此类模拟应对最强大的计算机的计算能力，因此，近似值中的计算错误始终是显着的，并且可能是压倒性的。在过去的二十年中，项目调查人员开发了一种系统的方法来产生对多物理系统模型近似解决方案误差的准确计算估计。在此项目中，研究人员探讨了从这种方法中使用错误估计的使用，以指导计算资源的有效使用，以最大程度地提高多物理系统的近似解决方案的忠诚度。他们还应用了误差估计，以准确量化基于模型的近似解决方案的多物理系统行为预测的不确定性。该项目的结果将增强国家工程师和科学家调查和预测对国家安全和基础设施重要的复杂物理系统的行为的能力。该项目解决了与使用复杂的尖端多欺诈化数字方法相关的关键问题，用于多尺度，多物理模型来追求科学推断和工程设计。该研究基于多物理学的后验误差分析，通过使用伴随问题和可计算的残差来量化各种离散化步骤的影响。该项目的主要重点是双重的：（1）开发和分析使用准确的错误估计来指导离散化选择的方法，以便以大致最小的计算成本达到所需的准确性；（2）研究如何扩展准确的误差估计方法以解决多物理系统的不确定性定量，其中“离散化”包括对随机过程的采样，总体误差是离散化和采样误差的组合。研究人员追求开发新型的多阶段方法，以构建有效的数值解决方案，并将后验错误分析扩展到统计计算。该项目还涉及将后验错误分析理论扩展到双曲线问题和非标准量。